GIBBS-WEIGHTED K-MEANS SEGMENTATION APPROACH

WITH INTENSITY INHOMOGENEITY CORRECTION

Chia-Yen Lee

, Chiun-Sheng Huang

, Yeun-Chung Chang

, Yi-Hong Chou

and Chung-Ming Chen

Institute of Biomedical Engineering, National Taiwan University, Taipei, Taiwan

Department of Surgery, National Taiwan University Hospital, Taipei, Taiwan

Department of Medical Imaging, National Taiwan University Hospital, Taipei, Taiwan

Department of Radiology, Taipei Veterans General Hospital, Taipei, Taiwan

Keywords: Breast Sonogram, Intensity Inhomogeneity Correction, Segmentation, Fuzzy Cell Competition, Clustering.

Abstract: Intensity inhomogeneity caused by an ultrasonic attenuation beam within the body results in an artifact

effect. It frequently degrades the boundary and texture information of a lesion in a breast sonogram. A new

Gibbs-weighted K-means segmentation approach with intensity inhomogeneity correction is proposed to

cluster the prominent components provided by fuzzy cell competition algorithm for segmenting lesion

boundaries automatically with reducing the influence of the intensity inhomogeneity. The information of

fuzzy C-means, normalized cut, and cell-based fuzzy cell competition algorithm are combined as the feature

vector for cell-based clustering. 49 breast sonograms with intensity inhomogeneity, each from a different

subject, are randomly selected for performance analysis. The mean distance between the lesion boundaries

attained by the proposed algorithm and the corresponding manually delineated boundaries defined by two

radiologists is 1.571±0.513 pixels. (Assessing Chan and Vese level set method for intensity inhomogeneity-

correction segmentation in the same way, the mean distance error is3.299±1.203 pixels, for the 49 images.)

The results show that Gibbs-weighted K-means segmentation approach with intensity inhomogeneity

correction could not only correct the intensity inhomogeneity effect but also improve the segmentation

results.

1 INTRODUCTION

Boundary segmentation is an essential step for the

quantitative analysis of sonographic breast lesions.

The shape and contour of the lesion are important

indicators for the malignancy of breast lesions.

Varieties of approaches have been proposed for

segmentation of sonographic breast lesions. Despite

the satisfactory performances that have been

repeatedly reported, each class of algorithms suffer

certain types of fundamental deficiency and

boundary delineation of sonographic breast lesions

remains as a hard task in general. For example, for a

breast lesion with a complex texture pattern in the

vicinity of the lesion boundary, the deformation of

the deformable models (Chen, 2003) and level-set

methods (Chan, 2001) is easily blocked by the

textural structures or leaks out from the weak edges

(Chen, 2005).

A cell-based approach, Gibbs-weighted K-means

segmentation approach with intensity inhomogeneity

correction, is proposed instead of the conventional

pixel-based approaches. Based on cell-based concept,

one important part of proposed approach, fuzzy cell

competition algorithm, is applied to identify all

prominent components in an ROI simultaneously. A

prominent component is the substructure of a tissue

or a part of breast lesion with a visually perceivable

boundary. In general, each breast lesion is composed

of a very limited number of prominent components,

with which the lesion boundary can be easily

derived by a boundary delineation approach, e.g.,

cell-based clustering.

In that way, the Gibbs-weighted K-means

segmentation approach with intensity inhomogeneity

correction combines the information of each

prominent component computed by fuzzy c-means,

normalized cut, and

cell-based fuzzy cell

competition algorithm as the feature vector for cell-

based clutstering. However, intensity inhomogeneity,

which may be composed of acoustic shadow and

enhancement, is a common artifact in a breast

381

Lee C., Huang C., Chang Y., Chou Y. and Chen C..

GIBBS-WEIGHTED K-MEANS SEGMENTATION APPROACH WITH INTENSITY INHOMOGENEITY CORRECTION.

DOI: 10.5220/0003946803810384

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 381-384

ISBN: 978-989-8565-03-7

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

sonogram. Intensity inhomogeneity may also

degrade the boundary and texture information. The

typical example is the shadowing artifact may result

in weak edges or missing edges, aggravating the

difficulty of extracting the lesion boundaries for

further lesion characterization.

In a breast sonogram, an ultrasonic beam may

travel through several different tissues, such as fat,

muscle, mammary gland, lesion, and so on.

Moreover, the composition of tissue types along the

traversing path varies with ultrasonic beams. It

suggests that the attenuation effect is basically

spatially-variant, which is not only a function of

traveling distance but also a function of ultrasonic

beam. To reduce the influence of intensity

inhomogeneity in breast sonograms, polynomial

surface model (Lee, 2010) is applied to take into

account the spatially-variant nature of the

attenuation and minimize the probability of being

trapped in a local minimum.

2 MATERIALS AND METHODS

The ROI is assumed to be composed of the

foreground and background regions. Each region is

assumed to be a homogeneous area with a mean

intensity,







BF,

, where F and B denote the

foreground and background regions, respectively.

Let

g denote the observed gray level of the ith

pixel in the ROI. Then,

g may be expressed as

iii

ng 









(1)

where



BF, depending on the location of the

ith pixel. The last two terms model the intensity

inhomogeneity and the noise. The noise is assumed

to be normally-distributed with zero mean. The

intensity inhomogeneity in an ROI is modeled as a

spatially-variant normal distribution with a constant

variance and spatially-variant means, which forms a

polynomial surface of order n denoted by



in Eq.

(1). The last term

n is the composition of the noise

and the variation on the polynomial surface of the

intensity inhomogeneity, which is also normally-

distributed.

The proposed segmentation with intensity

inhomogeneity correction scheme is formulated as

an EM algorithm composed of two major parts,

namely, Gibbs-weighted K-means segmentation

algorithm for the E-step as well as inhomogeneity

estimation and correction for the M-step. In the E-

step, the Gibbs-weighted K-means segmentation

algorithm divides the ROI into reasonable

foreground and background regions by the new

Gibbs-weighted K-means segmentation algorithm

taking the outputs of the fuzzy c-means, fuzzy cell-

competition algorithm and normalized cut algorithm

as the feature vectors. In the M-step, the nth order

polynomial surface of the inhomogeneity field is

estimated by using the least squared fitting. The

iteration process terminates when the difference of

the estimated fields derived in two consecutive

iterations is stabilized.

2.1 E Step

Based on the intensity inhomogeneity derived in the

previous M-step, the E-step aims to provide a

segmentation estimation of the foreground and

background regions close to the true partition as

possible for the M-step by the Gibbs-weighted K-

means segmentation algorithm. The proposed

algorithm comprises three components: fuzzy cell

competition (FCC), normalized cut (NC), fuzzy c

mens (FCM).

2.1.1 Fuzzy Cell Competition

The goal of the FCC algorithm is to derive a

minimum number of prominent components

constituting the breast lesion in a breast sonogram.

The FCC algorithm (Lee, 2010) is a cell-based

segmentation algorithm designed to find all

prominent components in an ROI, the boundaries of

which coincide with the visually perceived

boundaries.

The cost function of the FCC algorithm

characterizes the overall regional homogeneity of all

regions and the total boundary strength of all

boundary segments, which are defined by the first

and second terms, respectively, in Eq. (2) for the ith

iteration with





logmax





















)log(

)(









(2)

and































)()()()()(

hjk

rcccfur



(3)

subject to

1





(4)

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

382

where

 : the set of regions in the ith iteration;

n : the number of pixels enclosed by

r ;

c : the h

cell of

r in the ith iteration;

)(

hjk

cf : the gray level of the k

pixel enclosed by

c ;

)(



: the mean gray levels of the pixels

)(



: enclosed by

c and

r , respectively;

n : the number of pixels within the ROI;



: the variance of the gray levels of the

pixels within the ROI;

: the membership of cell j to region

r ;

 : the set of boundary segments on the

boundaries identified by the second pass

of watershed transformation;

b : the j

boundary segment.

A boundary segment is a small portion of a regional

boundary uniquely shared by two elementary cells

residing within two different regions. The boundary

strength of a boundary segment, denoted by

p , is a

measure of the statistical significance of difference

between these two elemental cells. It is defined as

the minimum of the p-values of three two-tailed

Kolmogorov-Smirnov (KS) tests respectively

comparing the gray level distributions of pixels in

two elementary cells and those on the boundary

segment. The probability of

b being an edge is

defined as

)1(

p

. Moreover, the overall boundary

strength of the regional contour



C defined by a

fuzzy class labeling

 is defined as the joint

probability of all boundary segments on



C being

edges, i.e.,











p )1( . The membership

function in each iteration may be derived as:





)1(

)()()()(









































hjk

rcccf



(5)

2.1.2 Gibbs-weighted K-means

Segmentation Algorithm

K-means aims to divide a dataset into several

groups. It needs feature vectors as inputs and a cost

function to define the quality of the partition result.

The Gibbs model is generally used to site labeling

problem to build the spatial dependency of each

neighboring site, i.e., pixel, edges, elements, and

region (Mohamed, 2004) via the set of clique

potentials. A Gibbs-weighted K-means is proposed

to provide a more general and flexible method to

make to 2D sonogram segmentation results better.

The basic idea is minimizing a cost function shown

as Eq. (6) which the sum of squares of the distances

of each cell to the two cluster center, i.e.,

background and foreground. For the pre-defined

cluster number K, the cost function is defined by

 

kCC

CCGS









(6)

where the Gibbs-weighted is as following





UzG





exp

(7)

and the energy function is a sum of clique potentials

V over the set of all adjacent pairs of cells





lVU









(8)

The clique potential is with respect to its

neighboring cell to be defined as:





hkhC







(9)

where













llif





(10)

There are six parameters as the feature vector to be

inputs. The first two parameters are the mean gray-

level of each cell which obtained by fuzzy cell

competition, which are cell-based features. The third

and fourth parameters are the mean of membership

function of each cell. The membership functions of

each pixel is derived by fuzzy c means originally,

and then transfer to cell-based features, i.e., the

membership functions of each cell, via the cell

information obtained by the fuzzy cell competition

algorithm. The fifth and sixth parameters are the

mean of the second smallest eigenvalue of each cell

which derived by normalized cut algorithm via

combing the cell information.

2.2 M Step

The aim of this step is to fit the intensity

inhomogeneity field to a polynomial surface in a

bipartite ROI based on the model given in Eq. (1). It

is assumed that the mean of intensity inhomogeneity

GIBBS-WEIGHTED K-MEANS SEGMENTATION APPROACH WITH INTENSITY INHOMOGENEITY

CORRECTION

383

 of pixel i, i , may be modeled as a polynomial

surface. To estimate the polynomial surface, a least

squared fitting is employed to minimize the cost

function:













(11)

where



 if pixel i is in the foreground region

and



 if pixel i is in the background region.

Moreover,

),(

iii

yxf , where ),( yxf is a

polynomial function of order n. In this study, n is

set to 6.

3 RESULTS AND CONCLUSIONS

Forty-nine breast sonograms with intensity

inhomogeneity acquired from 1996 – 2004, each

from a different subject, were randomly selected

from the ultrasound image database in a teaching

hospital in Taiwan. The assessment is performed

based on the mean manually-delineated boundary of

each lesion in a breast sonogram. Each lesion was

demarcated by two graduate students and confirmed

by two experienced radiologists with 10- and 28-

year experience, respectively.

The mean distance between the lesion boundaries

attained by the proposed algorithm and the

corresponding manually delineated boundaries

defined by two radiologists is 1.571±0.513 pixels.

(Assessing Chan and Vese level set method for

intensity inhomogeneity-correction segmentation in

the same way, the mean distance error is

3.299±1.203 pixels, for the 49 images).

Figure 1 results of the proposed segmentation

approach and Chan and Vese level set method

applied on the intensity inhomogeneity image. Fig. 1

(a) shows the mean manually-delineated boundary

of the sonographic breast lesion given in Fig. 1 (c),

i.e., the original image.

Fig. 1(b) gives the intensity inhomogeneity field

and Fig. 1(f) shows the inhomogeneity-corrected

breast sonogram. The weak edge at the upper-right

portion of the lesion boundary becomes manifested

in the inhomogeneity-corrected breast sonogram.

The boundary derived by the Chan and Vese level

set method of the intensity inhomogeneity image and

intensity inhomogeneity-corrected image are shown

as Fig. 1 (e) and Fig. 1 (h), respectively. The Chan

and Vese level set method fails to capture the upper-

right portion of the lesion boundary which is a weak

edge caused by the intensity inhomogeneity. Fig. 1

(d) and Fig. 1 (g) show the cells (i.e., the prominent

components) derived by the FCC algorithm in the

intensity inhomogeneity image and intensity

inhomogeneity-corrected image, respectively. The

cell structures play the role of offering the feature

vectors to Gibbs-weighted K-means segmentation

algorithm, and the clustering result is shown as Fig.

1 (i). The boundary derived by Gibbs-weighted K-

means segmentation algorithm is shown as Fig. 1 (j).

(a) (b)

(f) (g) (h)

(i) (j)

Figure 1: The proposed algorithm segmentation results

compare with Chan and Vese level set method.

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Chan, T. F. and Vese, L. A., 2001. Active contours

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ultrasound images. Ultrasound in Medicine and

Biology 31(12).

Chen, D. R., Chang, R. F., Wu, W. J., Moon, M. K. and

Wu, W. L., 2003. 3-D breast ultrasound segmentation

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Lee, C. Y., Chou, Y. H., Huang, C. S., Chang, Y. C., Tiu,

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